The automaton transformation of infinite words over alphabet Fp = {0, 1, . . . , p− 1}, where p is a prime number, coincide with the continuous transformation (with respect to the p-adic metric) of a ring Zp of p-adic integers. The objects of the study are non-Archimedean dynamical systems generated by automata mappings on the space Zp. Measure-preservation (with the respect to the Haar measure...

If m is Haar measure in G (normalized so that m(G) = 1) we consider the set function m'{E) = m{E). (E is the set of all x, # £ £ . ) Since m is a measure on G possessing all defining properties of m it follows from the uniqueness of Haar measure that ?n'(E) =m(E) for every measurable set E. In other words a is a measure preserving transformation of G; the purpose of this note is to investigate ...

We fix a logical connection (Stone Pred : Set op −→ BA given by 2 as a schizophrenic object) and study coalgebraic modal logic that is induced by a functor T : Set −→ Set which is finitary, standard, preserves weak pullbacks and finite sets. We prove that for any such T , the cover modality nabla is a left (and its dual delta is a right) adjoint relative to Pω. We then consider monotone unary m...

We give new formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay [Europ. J. Combin. 4 149–160] for regular graphs. The general answer involves a quantity for infinite graphs that we call ‘tree entropy’, which we show is a logarithm of a normalized determinant of the graph Laplacian for infinite graphs. Tree entropy is also exp...

This paper proves some results concerning the polar factorisation of an integrable vector-valued function u into the composition u = u ◦ s, where u = ∇ψ almost everywhere for some convex function ψ, and s is a measure-preserving mapping. Not every integrable function has a polar factorisation; we extend the class of counterexamples. We introduce a generalisation: u has a polar inclusion if u(x)...